Modern Physics: Chapter 7 – Introduction to Wavefunctions

XKCD/849

So I already mentioned that everything has a matter-wave associated with it. Developing the idea a bit further, we get the concept of a wavefunction which, by convention, is denoted by \psi.

Each physical system is described by a state function which determines all can be known about the system. … The wavefunction has to be finite and single valued throughout position space, and furthermore, it must also be a continuous and continuously differentiable function.

– Derivation of the postulates of quantum mechanics from the first principles of scale relativity

Furthermore, a wavefunction is generally a complex function i.e it consists of both a real part and an imaginary part.

  • \psi (x, t) – A wavefunction

To get the probably of a particle being at a position x, you have to get a real value out of the complex wavefunction. Hence the probably of finding a particle described by a wavefunction \psi is the product of the function \psi at x and its complex conjugate \psi^{*} at x.

  • \psi (x_{0}) \psi^{*}(x_{0}) – probability of finding the particle at x_{0}.

To get the probability from a range x \in (x_{0}, x_{1}) where x is a continous variable, you can take the integral of \psi^{*}\psi from x_{0} to x_{1}:

  • P(x) = \int_{x_{0}}^{x_{1}} \psi^{*}(x) \psi(x) dx
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